Atomic size, measurement

The size assigned to an atom or ion requires a decision about where an atom stops. From quantum mechanics we learn that an atom has no sharp boundaries or surfaces. Nevertheless, chemists find it convenient to assign sizes to atoms according to the observed distances between atoms. Thus, atomic size is defined operationally—it is determined by measuring the distance between atoms. 

By measuring the partial molar heat of solution as a function of temperature for infinitely dilute concentrations of Cu, Ag, and Au in liquid tin, Oriani and Murphy51 have determined ACp for the liquid solutes to be 1.0, 0, and 3.0 cal/deg mole respectively. These numbers bear no relationship to the sign of the heat of solution, or to atom-size disparity, but seem to be related to the deviation from unity of the ratio of the masses of the components. 

Scientists routineiy study objects whose sizes extend far beyond the narrow range encountered in daiiy iife. Physicists, for exampie, study atomic nuciei measuring 10 m across, and astronomers study our universe, which spans about 10 m. Chemists are most often interested in matter on the smaiier side of this range. Length measurements in the iaboratory vary from meters to subatomic sizes, 10 m. To further simpiify the use of very iarge and very smaii numbers, scientists use prefixes that change the unit sizes by muitipies of 10. For instance. 

Different units are useful for measuring different-sized items. Eggs are sold by the dozen, paper is sold by the ream, and atoms are measured by the mole. Twenty-four pre-1982 pennies contain 1 mol of copper atoms. 

To accommodate this problem, scientists have come up with several approaches to measuring atomic sizes. A common one is called the covalent radius, which is half the distance between the nuclei of two identical atoms. This technique works well for atoms such as hydrogen or oxygen, both of which readily pair up to form and O2. But how would one determine the covalent radius of a noble gas, which exists only as single atoms ... 

As described before, the pore size of porous material ranges widely from atomic size to millimeter order. Different pore sizes are required for different applications of porous materials. Most porous materials do not have uniform pores. Pore size distribution is also an important property. Narrow pore size distribution, i.e., uniform pore size, is required for instance for filters and bioreactor beds. Mercury porosimetry and gas adsorption methods are commonly used to measure pores size and pores distribution. 

Zhou X-S, Wei Y-M, Liu L, Chen Z-B, Tang J, Mao B-W (2008) Extending the capability of STM break junction for conductance measurement of atomic-size nanowires an electrochemical strategy. J Am Chem Soc 130 13228-13230... 

The first problem was resolved when it was shown that the Es values for symmetric groups are a linear function of van der Waals radii42. The latter have long been held to be an effective measure of atomic size. The second and third problems were solved by... 

The first problem was resolved when it was shown that the Es values for symmetric groups are a linear function of van der Waals radii41. The latter have long been held to be an effective measure of atomic size. The second and third problems were solved by Charton, who proposed the use of the van der Waals radius as a steric parameter42 and developed a method for the calculation of group van der Waals radii for tetracoordinate symmetric top substituents MZ3 such as the methyl and trifluoromethyl groups43. In later work the hydrogen atom was chosen as the reference substituent and the steric parameter v was defined as ... 

The studies on the performance of effervescent atomizer have been very limited as compared to those described above. However, the results of droplet size measurements made by Lefebvre et al.t87] for the effervescent atomizer provided insightful information about the effects of process parameters on droplet size. Their analysis of the experimental data suggested that the atomization quality by the effervescent atomizer is generally quite high. Better atomization may be achieved by generating small bubbles. Droplet size distribution may follow the Rosin-Rammler distribution pattern with the parameter q ranging from 1 to 2 for a gas to liquid ratio up to 0.2, and a liquid injection pressure from 34.5 to 345 kPa. The mean droplet size decreases with an increase in the gas to liquid ratio and/or liquid injection pressure. Any factor that tends to impair atomization quality, and increase the mean droplet size (for example, decreasing gas to liquid ratio and/or injection pressure) also leads to a more mono-disperse spray. 

The methodology for obtaining the partial atomic volume and its application as a realistic measure of atomic size in metals and alloys has been discussed by Bhatia and Cahn (2005) they illustrated its use as a powerful tool in understanding the behaviour of solid solutions in both ordered and disordered states. 

Hartree-Fock calculations of the three leading coefficients in the MacLaurin expansion, Eq. (5.40), have been made [187,232] for all atoms in the periodic table. The calculations [187] showed that 93% of rio(O) comes from the outermost s orbital, and that IIo(O) behaves as a measure of atomic size. Similarly, 95% of IIq(O) comes from the outermost s and p orbitals. The sign of IIq(O) depends on the relative number of electrons in the outermost s and p orbitals, which make negative and positive contributions, respectively. Clearly, the coefficients of the MacLaurin expansion are excellent probes of the valence orbitals. The curvature riQ(O) is a surprisingly powerful predictor of the global behavior of IIo(p). A positive IIq(O) indicates a type 11 momentum density, whereas a negative rio(O) indicates that IIo(O) is of either type 1 or 111 [187,230]. MacDougall has speculated on the connection between IIq(O) and superconductivity [233]. 

Average size measured by SEM, t>Estimated from surface area, Measured by BET method, Atomic ratio of Si to Vg rf, where Vg rf is the amount of surface vanadium atoms. 

An attempt has also been made to derive the binding energy of atoms in clusters from a measurement of the critical energy deficit of cluster ions. For n+ cluster ions of m atom size, from consideration of Born-Haber energy cycle, the critical ion energy deficit can be easily shown to be given by100... 

Zeolites (3 were treated with a NaBO, solution, and the porous properties of boronated samples were investigated by sorption measurements with benzene and nitrogen as adsorbate, TEM, SEM and composition analysis. It is shown that the micropores are converted into the mesopores and the mesopores are developed into larger mesopores due to the extraction of framework silicon by base. The small atom size of boron and the poor stability of boron in framework should be responsible for the silicon removal in a large amount. The dissolution of silicon also causes the corrosion of outer surface of particles and the decrease of particle size. 

Figure 10.6 Tracer diffusivities in glassy NisoZrso of various solute atoms as a function of their size (as measured by their metallic radii) [25]. Reprinted, by permission, from H. Hahn and R.S. Averback, "Dependence of tracer diffusion on atomic size in amorphous Ni-Zr," Phys. Rev. B, Vol. 37, p. 6534. Copyright 1988 by the American Physical Society.

The classical theory of Hume-Rothery states that a difference in atomic diameters of solute and solvent atoms of more than 15% produces restricted solid solubility. The closest distance of approach of the atoms in the crystals of the element is taken as a measure of the atomic size. Substitution of a larger atom into a lattice requires a high amount of energy due to the concomitant disorganization of the parent lattice. However, the size factor becomes less important [221] when the difference in size is 8% or less. It is desirable (though not essential) that the size factor and the crystal structure of the elements producing a solid solution in all proportions be favourable. It is, however, apparent that if elements forming alloys did not possess the same crystal structure, a continuous series of solid solutions would be impossible. 

---